I'm wondering how the total energy of a combined Earth-rocket system is conserved in a rocket launch?
Before launch, kinetic and gravitational potential energy of the rocket are 0. It then launches, and both KE and PE start to increase. How does this obey conservation of energy? Is it because the rocket's mass decreases due to the ejection of propellant?
It then launches, and both KE and PE start to increase. How does this obey conservation of energy?
You forgot thermal energy and chemical potential energy, etc. Thermal energy also increases, but chemical potential energy decreases dramatically. Indeed, that is the primary characteristic of a good rocket fuel.
For a chemical rocket, the energy source is chemical energy. This energy becomes thermal and kinetic energy of the ejected propellant and of the rocket itself. As the rocket moves upward from the pad, some of its kinetic energy becomes gravitational potential energy between the rocket and the Earth.
If the rocket's speed exceeds the exhaust velocity of the propellant, which it may in the later part of its flight, the propellant loses kinetic energy when ejected, but total energy is still conserved.
Energy conservation is a difficult path for understanding rockets: it's better to focus on momentum.
